One of the remaining outstanding issues, concerning the study of the Einstein equations in the presence of a timelike boundary, is the geometric uniqueness problem of Friedrich [1]}. Apart from the construction of asymptotically Anti-de-Sitter spacetimes [2]}, where the timelike boundary is a conformal boundary at spacelike infinity and our recent work in the totally geodesic case, [3]}, all results establishing well-posedness, for some formulations of the IBVP, impose certain gauge conditions on the boundary, and the boundary data depend on these choices. In particular, given a solution to the Einstein equations with a timelike boundary, different gauge choices will lead to different boundary data, in each of the formulations for which well-posedness is known.
